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Definition df-trkg 26253
Description: Define the class of Tarski geometries. A Tarski geometry is a set of points, equipped with a betweenness relation (denoting that a point lies on a line segment between two other points) and a congruence relation (denoting equality of line segment lengths). Here, we are using the following:
  • for congruence, (𝑥 𝑦) = (𝑧 𝑤) where = (dist‘𝑊)
  • for betweenness, 𝑦 ∈ (𝑥𝐼𝑧), where 𝐼 = (Itv‘𝑊)
With this definition, the axiom A2 is actually equivalent to the transitivity of addition, eqtrd 2859.

Tarski originally had more axioms, but later reduced his list to 11:

  • A1 A kind of reflexivity for the congruence relation (TarskiGC)
  • A2 Transitivity for the congruence relation (TarskiGC)
  • A3 Identity for the congruence relation (TarskiGC)
  • A4 Axiom of segment construction (TarskiGCB)
  • A5 5-segment axiom (TarskiGCB)
  • A6 Identity for the betweenness relation (TarskiGB)
  • A7 Axiom of Pasch (TarskiGB)
  • A8 Lower dimension axiom (DimTarskiG≥‘2)
  • A9 Upper dimension axiom (V ∖ (DimTarskiG≥‘3))
  • A10 Euclid's axiom (TarskiGE)
  • A11 Axiom of continuity (TarskiGB)
Our definition is split into 5 parts:
  • congruence axioms TarskiGC (which metric spaces fulfill)
  • betweenness axioms TarskiGB
  • congruence and betweenness axioms TarskiGCB
  • upper and lower dimension axioms DimTarskiG
  • axiom of Euclid / parallel postulate TarskiGE

So our definition of a Tarskian Geometry includes the 3 axioms for the quaternary congruence relation (A1, A2, A3), the 3 axioms for the ternary betweenness relation (A6, A7, A11), and the 2 axioms of compatibility of the congruence and the betweenness relations (A4,A5).

It does not include Euclid's axiom A10, nor the 2-dimensional axioms A8 (Lower dimension axiom) and A9 (Upper dimension axiom) so the number of dimensions of the geometry it formalizes is not constrained.

Considering A2 as one of the 3 axioms for the quaternary congruence relation is somewhat conventional, because the transitivity of the congruence relation is automatically given by our choice to take the distance as this congruence relation in our definition of Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.) (Revised by Thierry Arnoux, 27-Apr-2019.)

Assertion
Ref Expression
df-trkg TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
Distinct variable group:   𝑓,𝑝,𝑖,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-trkg
StepHypRef Expression
1 cstrkg 26230 . 2 class TarskiG
2 cstrkgc 26231 . . . 4 class TarskiGC
3 cstrkgb 26232 . . . 4 class TarskiGB
42, 3cin 3918 . . 3 class (TarskiGC ∩ TarskiGB)
5 cstrkgcb 26233 . . . 4 class TarskiGCB
6 vf . . . . . . . . . 10 setvar 𝑓
76cv 1537 . . . . . . . . 9 class 𝑓
8 clng 26237 . . . . . . . . 9 class LineG
97, 8cfv 6343 . . . . . . . 8 class (LineG‘𝑓)
10 vx . . . . . . . . 9 setvar 𝑥
11 vy . . . . . . . . 9 setvar 𝑦
12 vp . . . . . . . . . 10 setvar 𝑝
1312cv 1537 . . . . . . . . 9 class 𝑝
1410cv 1537 . . . . . . . . . . 11 class 𝑥
1514csn 4550 . . . . . . . . . 10 class {𝑥}
1613, 15cdif 3916 . . . . . . . . 9 class (𝑝 ∖ {𝑥})
17 vz . . . . . . . . . . . . 13 setvar 𝑧
1817cv 1537 . . . . . . . . . . . 12 class 𝑧
1911cv 1537 . . . . . . . . . . . . 13 class 𝑦
20 vi . . . . . . . . . . . . . 14 setvar 𝑖
2120cv 1537 . . . . . . . . . . . . 13 class 𝑖
2214, 19, 21co 7149 . . . . . . . . . . . 12 class (𝑥𝑖𝑦)
2318, 22wcel 2115 . . . . . . . . . . 11 wff 𝑧 ∈ (𝑥𝑖𝑦)
2418, 19, 21co 7149 . . . . . . . . . . . 12 class (𝑧𝑖𝑦)
2514, 24wcel 2115 . . . . . . . . . . 11 wff 𝑥 ∈ (𝑧𝑖𝑦)
2614, 18, 21co 7149 . . . . . . . . . . . 12 class (𝑥𝑖𝑧)
2719, 26wcel 2115 . . . . . . . . . . 11 wff 𝑦 ∈ (𝑥𝑖𝑧)
2823, 25, 27w3o 1083 . . . . . . . . . 10 wff (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))
2928, 17, 13crab 3137 . . . . . . . . 9 class {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}
3010, 11, 13, 16, 29cmpo 7151 . . . . . . . 8 class (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
319, 30wceq 1538 . . . . . . 7 wff (LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
32 citv 26236 . . . . . . . 8 class Itv
337, 32cfv 6343 . . . . . . 7 class (Itv‘𝑓)
3431, 20, 33wsbc 3758 . . . . . 6 wff [(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
35 cbs 16483 . . . . . . 7 class Base
367, 35cfv 6343 . . . . . 6 class (Base‘𝑓)
3734, 12, 36wsbc 3758 . . . . 5 wff [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
3837, 6cab 2802 . . . 4 class {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}
395, 38cin 3918 . . 3 class (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})
404, 39cin 3918 . 2 class ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
411, 40wceq 1538 1 wff TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
Colors of variables: wff setvar class
This definition is referenced by:  axtgcgrrflx  26262  axtgcgrid  26263  axtgsegcon  26264  axtg5seg  26265  axtgbtwnid  26266  axtgpasch  26267  axtgcont1  26268  tglng  26346  f1otrg  26671  eengtrkg  26786
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